Generators of Quantum Markov Semigroups

TL;DR

This paper investigates the structure of generators of non-uniformly continuous quantum Markov semigroups acting on bounded operators, extending classical results and providing new insights into their forms.

Contribution

It extends the characterization of QMS generators beyond the uniformly continuous case, aligning with Lindblad and Stinespring's results and exploring unbounded operators.

Findings

Generators have forms reflecting Lindblad and Stinespring theorems.

Progress towards forms reflecting Kraus' result.

Examples verify dense domains of unbounded operators.

Abstract

Quantum Markov Semigroups (QMSs) originally arose in the study of the evolutions of irreversible open quantum systems. Mathematically, they are a generalization of classical Markov semigroups where the underlying function space is replaced by a non-commutative operator algebra. In the case when the QMS is uniformly continuous, theorems due to Lindblad \cite{lindblad}, Stinespring \cite{stinespring}, and Kraus \cite{kraus} imply that the generator of the semigroup has the form $$L(A)=\sum_{n=1}^{\infty}V_n^*AV_n +GA+AG^*$$ where $V_n$ and G are elements of the underlying operator algebra. In the present paper we investigate the form of the generators of QMSs which are not necessarily uniformly continuous and act on the bounded operators of a Hilbert space. We prove that the generators of such semigroups have forms that reflect the results of Lindblad and Stinespring. We also make some progress towards forms reflecting Kraus' result. Lastly we look at several examples to clarify our findings and verify that some of the unbounded operators we are using have dense domains.